This paper revisits the simple, but empirically salient, problem of inference on a real-valued parameter that is partially identiﬁed through upper and lower bounds with asymptotically normal estimators. A simple conﬁdence interval is proposed and is shown to have the following properties:
- It is never empty or awkwardly short, including when the sample analog of the identiﬁed set is empty.
- It is valid for a well-deﬁned pseudotrue parameter whether or not the model is well-speciﬁed.
- It involves no tuning parameters and minimal computation.
Computing the interval requires concentrating out one scalar nuisance parameter. In most cases, the practical result will be simple: To achieve 95% coverage, report the union of a simple 90% (!) conﬁdence interval for the identiﬁed set and a standard 95% conﬁdence interval for the pseudotrue parameter.
For uncorrelated estimators –notably if bounds are estimated from distinct subsamples–and conventional coverage levels, validity of this simple procedure can be shown analytically. The case obtains in the motivating empirical application (de Quidt, Haushofer, and Roth, 2018), in which improvement over existing inference methods is demonstrated. More generally, simulations suggest that the novel conﬁdence interval has excellent length and size control. This is partly because, in anticipation of never being empty, the interval can be made shorter than conventional ones in relevant regions of sample space.